Econ/Math C103 – Final
12/18/2020
Instructions: This is a 24-hour open-book take-home exam. You can use any result from the lecture notes, problem sets, and problem set solutions.
The weight of each question is indicated next to it. Please write clearly and explain your answers. Make sure to upload a pdf file of your (scanned or typed-up) solutions to Gradescope by 9am PDT on Saturday, December 19.
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Please write the following honor code, sign below it, and include it with your solu- tions: “I swear on my honor that: (1) I have not used the internet in relation to this exam other than: accessing the materials in the bCourses website of C103 and uploading my answers to Gradescope, (2) I have neither given nor received aid on this exam.”
Good luck!
1. (20pts) Consider a marriage market with four men M = {1, 2, 3, 4} and five women
W = {a, b, c, d, e} whose preferences are given by:
R1 R2 R3 R4 bcec eabd 1b3e cdd4 deca a2ab
Ra Rb Rc Rd Re 31c23 242d4 43341 a241e 1b132
List all the stable matching(s). Explain why there are no other stable matchings.
2. (20pts) Consider the following two-player game where Player 1 chooses one of the four rows and Player 2 chooses one of the four columns:
C1 C2 C3 C4 R1 6,5 0,1 0,1 0,0 R2 2,0 1,0 1,1 1,0 R3 1,0 4,5 0,1 3,1 R4 0,0 0,0 4,0 2,1
(a) What are the strategies that survive IESDS? 1
(b) At each step of the elimination what were your rationality and knowledge assumptions?
(c) Find all the (possibly mixed) Nash equilibria.
3. (20pts) Let (N,X,R,μE) be a housing market, and let μ be a Pareto efficient assignment that is not necessarily in the core. Show that under the assignment μ, there exists an agent who receives his/her top object.
4. (20pts) Consider a quasi-linear model with n agents and m distinct indivisible objects O = {o1, . . . , om}. Each agent can consume at most one object and m < n. The set K consists of all one-to-one functions μ : O → N, where μ(ol) = i means that object ol is allocated to agent i at the “project choice” μ. Let Θi = [0,1] denote the set of possible valuations of agent i for object o1. We interpret objects with lower indices to be of higher quality: All agents prefer o1 to o2 to o3 etc. To modelthis,leta1,...,am ∈[0,1]besuchthat1=a1 >a2 >…>am >0,and:
alθi if ∃l ∈ {1,…,m} s.t. i = μ(ol)
vi(μ, θi) =
for all i ∈ N, μ ∈ K, and θi ∈ Θi. Derive the pivotal VCG mechanism.
5. (20pts) Consider the auction of a single indivisible object to two bidders whose valuations are distributed independently. Bidder 1’s valuation is distributed uni- formly on the interval [0, 1] (i.e., F1(x1) = x1 and f1(x1) = 1). Bidder 2’s valuation is distributed on the interval [2, 3] with a continuous nondecreasing density f2 such that f2 (2) ∈ (0, 21 ). Consider a revenue-maximizing feasible direct auction mech- anism. Is the object allocated to one of the two bidders with probability one? Is the object always allocated to the bidder with the highest valuation?
0 otherwise
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